f'(] ) =. 1. (1 point) Use the definition of the derivative (don't be f' (2) = . tempted to take shortcuts!) to find the derivative of the function f'(3) = f (x) = 9x +4vx. 4. (1 point) Let f(x) = x -8 . Evaluate the following limits. Then state the domain of the function and the domain of the derivative. lim f(x) - f(8) Note: Enter intervals using interval notation. * +8 x -8 lim f(x) - f(8) f' (x) = * +8+ x -8 Domain of f (x) = Thus the function f(x) is not differentiable at 8. 5. (1 point) Enter a T or an F in each answer space below Domain of f'(x) = to indicate whether the corresponding statement is true or false. You must get all of the answers correct to receive credit. 1. If lim f(x) = 2 and lim g(x) = 0, then lim If (x)/g(x)] x-+6" does not exist -510 - 2. If lim f(x) = 0 and lim g(x) = 2, then lim [f (x)/g(x)] does not exist 3. If lim [f(x)g(x)] exists, then the limit is f(6)g(6) 4. If lim f(x) = co and limg(x) = co, then lim[f (x) - 0.5 x-+6 * +6 g(x) ] = 0 x2 + 2x - 3 lim x- + 2x - 3 5. lim x-1x2+4x -5 lim-1x2+4x - 5 6. (1 point) According to Newton's Law of Cooling , the rate of change of an object's temperature is proportional to the dif- ference between the temperature of the the object and that of 2. (1 point) -10 the surrounding medium. The accompanying figure shows the graph of the temperature 7 (in degrees Fahrenheit) versus time Identify the graphs A (blue), B( red) and C (green) as the graphs t (in minutes) for a cup of coffee, with initial temperature 200 of a function and its derivatives: degrees Fahrenheit, that is allowed to cool in a room with a con- aph of the function stant temperature of 75 degrees Fahrenheit. is the graph of the function's first derivative is the graph of the function's second derivative 3. (1 point) Let f(x) = 2. Then the expression f(xth ) - f(x ) can be written in the form h x(xth) where A is a constant and A = Using your answer from above we have: Click on the image to see a larger graph. f'(x) = lim J(xth) - f(x) h-0 h (a) Estimate T when t = 10 minutes: Finally, find each of the following: (b) Estimate dT /dt when t = 10 minutes: (c) Use the results of parts (a) and (b) to estimate the value Newton's Law of Cooling can be expressed as = k(T - of k. To) ,where k is the constant of proportionality and To is the temperature of the surrounding medium. Value of k